The British government does this with small portions of tax income. On certain collection forms you can put essentially "I'd like my five quid to go to ____". This resulted in a net increase in people paying taxes.
We already do, in a way, through elected officials. That's what a representative democracy is about, isn't it? A direct democracy would mean each person could specifically choose where their taxes go, but it is hard to implement and maintain and might result in richer people having more say.
If It was up to society the allocation would be simply wrong. E.g.: Nobody would pay for politicians who get their salary from taxes; or for military; or probably would overpay shelters. So, lazy people will get lazier and strong would loose motivation. That's socialism, lol